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Mirrors > Home > MPE Home > Th. List > Mathboxes > brtxpsd | Structured version Visualization version GIF version |
Description: Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
brtxpsd.1 | ⊢ 𝐴 ∈ V |
brtxpsd.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brtxpsd | ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5040 | . . 3 ⊢ (𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V))) | |
2 | opex 5333 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
3 | 2 | elrn 5747 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V)) ↔ ∃𝑥 𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉) |
4 | brsymdif 5098 | . . . . . 6 ⊢ (𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉 ↔ ¬ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉)) | |
5 | brv 5341 | . . . . . . . . 9 ⊢ 𝑥V𝐴 | |
6 | vex 3402 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
7 | brtxpsd.1 | . . . . . . . . . 10 ⊢ 𝐴 ∈ V | |
8 | brtxpsd.2 | . . . . . . . . . 10 ⊢ 𝐵 ∈ V | |
9 | 6, 7, 8 | brtxp 33868 | . . . . . . . . 9 ⊢ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ (𝑥V𝐴 ∧ 𝑥 E 𝐵)) |
10 | 5, 9 | mpbiran 709 | . . . . . . . 8 ⊢ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥 E 𝐵) |
11 | 8 | epeli 5447 | . . . . . . . 8 ⊢ (𝑥 E 𝐵 ↔ 𝑥 ∈ 𝐵) |
12 | 10, 11 | bitri 278 | . . . . . . 7 ⊢ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥 ∈ 𝐵) |
13 | brv 5341 | . . . . . . . 8 ⊢ 𝑥V𝐵 | |
14 | 6, 7, 8 | brtxp 33868 | . . . . . . . 8 ⊢ (𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉 ↔ (𝑥𝑅𝐴 ∧ 𝑥V𝐵)) |
15 | 13, 14 | mpbiran2 710 | . . . . . . 7 ⊢ (𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉 ↔ 𝑥𝑅𝐴) |
16 | 12, 15 | bibi12i 343 | . . . . . 6 ⊢ ((𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉) ↔ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
17 | 4, 16 | xchbinx 337 | . . . . 5 ⊢ (𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉 ↔ ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
18 | 17 | exbii 1855 | . . . 4 ⊢ (∃𝑥 𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉 ↔ ∃𝑥 ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
19 | 3, 18 | bitri 278 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V)) ↔ ∃𝑥 ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
20 | exnal 1834 | . . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴) ↔ ¬ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) | |
21 | 1, 19, 20 | 3bitrri 301 | . 2 ⊢ (¬ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴) ↔ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵) |
22 | 21 | con1bii 360 | 1 ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∀wal 1541 ∃wex 1787 ∈ wcel 2112 Vcvv 3398 △ csymdif 4142 〈cop 4533 class class class wbr 5039 E cep 5444 ran crn 5537 ⊗ ctxp 33818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-symdif 4143 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-eprel 5445 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-1st 7739 df-2nd 7740 df-txp 33842 |
This theorem is referenced by: brtxpsd2 33883 |
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